Abstract
The convergent beam electron diffraction (CBED) patterns of twisted bilayer samples exhibit interference patterns in their CBED spots. Such interference patterns can be treated as off-axis holograms and the phase of the scattered waves, meaning the interlayer distance can be reconstructed. A detailed protocol of the reconstruction procedure is provided in this study. In addition, we derive an exact formula for reconstructing the interlayer distance from the recovered phase distribution, which takes into account the different chemical compositions of the individual monolayers. It is shown that one interference fringe in a CBED spot is sufficient to reconstruct the distance between the layers, which can be practical for imaging samples with a relatively small twist angle or when probing small sample regions. The quality of the reconstructed interlayer distance is studied as a function of the twist angle. At smaller twist angles, the reconstructed interlayer distance distribution is more precise and artefact free. At larger twist angles, artefacts due to the moiré structure appear in the reconstruction. A method for the reconstruction of the average interlayer distance is presented. As for resolution, the interlayer distance can be reconstructed by the holographic approach at an accuracy of ±0.5 Å, which is a few hundred times better than the intrinsic z-resolution of diffraction limited resolution, as expressed through the spread of the measured k-values. Moreover, we show that holographic CBED imaging can detect variations as small as 0.1 Å in the interlayer distance, though the quantitative reconstruction of such variations suffers from large errors.
Highlights
We derived a formula for recovering the interlayer distance from the phase distribution reconstructed from Convergent beam electron diffraction (CBED) pattern
We showed that surprisingly even one interference fringe in the interference pattern is sufficient to reconstruct the phase shift and with this, the interlayer distance
One interference fringe is observed in CBED spots when either the twist angle or the probed region is small
Summary
Convergent beam electron diffraction (CBED) [1,2,3] has been routinely utilized for studying parameters of thick crystals: thickness [4], lattice parameters [5,6,7], and crystallographic deformations [8, 9]. CBED performed on two-dimensional (2D) crystals and van der Waals structures [10, 11] produces patterns which require different interpretation than in the case of thick crystals [12,13,14]. A particular advantage of holographic CBED is possibility to obtain z-information from a single CBED pattern. Lateral or (x,y) atomic positions can be accessed at sub-Ångstrom resolution through scanning procedure by electron ptychography [15], the access to z-information is possible by cross-sectional transmission electron microscopy (TEM) imaging [16]. Holographic CBED approach allows access to z-atomic positions and the interlayer distance in bilayer (BL) systems from a single CBED pattern. To present a systematic study and demonstrate the performance of the technique at different parameters, we provide simulated examples
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